An Algorithmic Description - Design and Analysis of Learning Classifier Systems - page 183

the shape of the radial-basis matching function are shown in Fig. 8.2. This form
of matching function was chosen to demonstrate the possibility of matching by
probability.
Rather than declaring μ k and σ k directly, the matching parameters 0 ≤ a k ≤

l ) a k / 100 and
σ k =10 −b k / 10 ,where[ l, u ] is that range of the input x .Thus, a k determines the
centre of the classifier, where 0 and 100 specify the lower and higher end of x ,
respectively. σ k is given by b k such that 10 − 50

≤ b k ≤

50 determine μ k and σ k by μ k = l +( u

−

100 and 0

≤ σ k ≤ 1, and a low b k gives
a wide spread of the classifier matching function. A new classifier is initialised
by randomly choosing a k uniformly from [0 , 100), and b k uniformly from [0 , 50).
The two values are mutated by adding a sample from N (0 , 10) to a k ,anda

sample from N (0 , 5) to b k , but ensuring thereafter that they still conform to

0 50. The reason for operating on a k ,b k rather than

μ k ,σ k is that it simplifies the mutation operation by making it independent of the
range of x for μ k and allows for non-linearity with respect to σ k . Alternatively,
one could simply acquire the mutation operator that was used by Butz, Lanzi
and Wilson [52].

≤ a k ≤ 100 and 0 ≤ b k ≤

Matching by Soft Intervals

Matching by soft intervals is similar to the interval matching that was introduced
in XCS by Wilson [239], with the difference that here, the intervals have soft
boundaries. The reason for using soft rather than hard boundaries is to express
the fact that we are never absolutely certain about the exact location of these
boundaries, and to avoid the need to explicitly care about having each input
matched by at least one classifier.
To avoid the representational bias of the centre/spread representation of Wil-
son [239], the lower/upper bound representation that was introduced and ana-
lysed by Stone and Bull [203] is used instead. The softness of the boundary
is provided by an unnormalised Gaussian that is attached to both sides of the
interval within which the classifier matches with probability 1. To avoid the
boundaries from being too soft, they are partially included in the interval. More
precisely, when specifying the interval for classifier k by its lower bound l k and
upper bound u k , exactly one standard deviation of the Gaussian is to lie in-
side this interval, with the additional requirement of having 95% of the area
underneath the matching function inside this interval. More formally, we need

0 . 95( b k + √ 2 πσ k )= b k to hold to have the interval b k = u k −

l k specify 95%

of the area underneath the matching function, where b k gives the width of the

interval where the classifier matches with probability 1, using the area √ 2 πσ
underneath an unnormalised Gaussian with standard deviation σ .Therequire-
ment of the specified interval extending by one standard deviation to either side

of the Gaussian is satisfied by b k +0 . 6827 √ 2 πσ k = b k , based on the fact that
the area underneath the unnormalised Gaussian within one standard deviation